Both sentences are correct, the one using 'alike' just sounds a bit more intellectual because that word isn't used as often. 'Alike', as the other commenter said, is a slightly fancier word, most people will use a more simple word but there are set phrases that use 'alike'. "great minds think alike" "look/sound alike"

Yes, both are correct and mean the same thing. Using “both” is a little more common, using “alike” is a little more fancy.